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Quasi‐symmetric designs with fixed difference of block intersection numbers
Author(s) -
Pawale Rajendra M.
Publication year - 2007
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20106
Subject(s) - mathematics , combinatorics , intersection (aeronautics) , zero (linguistics) , complement (music) , block (permutation group theory) , discrete mathematics , philosophy , linguistics , biochemistry , chemistry , complementation , engineering , gene , phenotype , aerospace engineering
Abstract The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x , y and λ > 1 are proved. (1) Let D be a quasi‐symmetric design with z = y − x and v ≥ 2 k . If x ≥ 1 + z + z 3 then λ < x + 1 + z + z 3 . (2) Let D be a quasi‐symmetric design with intersection numbers x , y and y − x = 1. Then D is a design with parameters v = (1 + m ) (2 + m )/2, b = (2 + m ) (3 + m )/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2. (3) Let D be a triangle free quasi‐symmetric design with z = y − x and v ≥ 2 k , then x ≤ z + z 2 . (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x , y = x + z . (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x , y = x + 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007